We remark that no algorithm can solve the problem, uniformly in d and E, without assuming a compactness condition on the root space or coefficient space. We also note that given any polynomialf,P = uf(bz) E P,j(l) for appropriate constants a and 6. Hereafter, x satisfying Ix tjl 5 E will be called an c-root off. Our focus here is to minimize the branching (if-then, else) appearing in the algorithm. This minimum number is called the topological complexity, a concept which has been recently developed by Smale (1989) as a part of a new notion of a machine (or an algorithm). An algorithm is a function from input space BP to output space R” which allows only rational maps and branchings. It may be viewed as a directed tree with the following types of nodes; one root (input), leaves (output), computational nodes (rational maps), and branching nodes. Our algorithm has (d 1) branching nodes (d leaves) and hence establishes that the upper bound of topological complexity of the above problem P is no more than (d 1).
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